The \(t\)-Pebbling Conjecture on Products of Complete \(r\)-Partite Graphs

Abstract

The \(t\)-pebbling number \(f_t(G)\) of a graph \(G\) is the least positive integer \(m\) such that however these \(m\) pebbles are placed on the vertices of \(G\), we can move \(t\) pebbles to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. In this paper, we study the generalized Graham’s pebbling conjecture \(f_t(G \times H) \leq f(G)f_t(H)\) for the product of graphs when \(G\) is a complete \(r\)-partite graph and \(H\) has a \(2t\)-pebbling property.